Earlier this month Paul Krugman gave a speech titled Mr Keynes and the Moderns at a Cambridge University conference commemorating the 75th anniversary of the publication of The General Theory of Employment, Interest and Money.

I am glad to note that he has therein seen fit to address a most curious absence in Keynes (see John Maynard Keynes and the dog that did not bark. Although the Great Depression that began in 1929 was accompanied by the wholesale failure of hundreds of banks, the General Theory published in 1936 did not have a word to say about bank rescues. I suggested that this was not a casual oversight on the part of Keynes but proof not only that he never understood the significance of banks but also that most of his assertions assumed an economy where money consisted entirely of currency and would collapse when banks were introduced into the equation.

Krugman's explanation for the absence of banks in Keynes is half-hearted, as if he himself is not really convinced:

"Perhaps the most surprising omission in the General Theory -- and the one that has so far generated the most soul-searching among those macroeconomists who had not forgotten basic Keynesian concepts -- is the book's failure to discuss banking crises. There's basically no financial sector in the General Theory; macroeconomics ever since has more or less discussed money and banking off to the side, giving it no central role in business cycle analysis.

"I'd be curious to hear what Keynes scholars have to say about this omission. Keynes was certainly aware of the possibility of banking problems; his 1931 essay 'The Consequences to the Banks of the Collapse of Money Values' is a razor-sharp analysis of just how deflation could produce a banking crisis, as indeed it did in the United States.

"But not in Britain, which may be one reason Keynes left the subject out of the General Theory. Beyond that Keynes was -- or at least that's how it seems to a Part 1er -- primarily concerned with freeing minds from Say's Law and the notion that, if there was any demand problem, it could be solved simply by increasing the money supply. A prolonged focus on banking issues could have distracted from that central point."

I am afraid Krugman doesn't get it. Even if Keynes had mentioned banks on every page of the General Theory it still wouldn't change the basic fact that Keynes's model is one in which money exists only in the form of currency and that it collapses once banks are introduced.

In what follows I shall demonstrate that every important assertion in Krugman's speech (and by implication in Keynes's theory) assumes an economy in which money consists entirely of currency and that each of them collapses when banks are introduced.

This is what Krugman says in his speech:

"… Keynes pointed out … that the schedules showing the supply and demand for funds can only be drawn on the assumption of a given level of income. Allow for the possibility of a rise in income, and you get Figure 3 -- which is Keynes's own figure, and a horrible drawing it is: [I shall take Krugman at his word and drop the figure]

"Figure 4 is my own version, very much along Hicks's lines: we imagine that a rise in GDP shifts the savings schedule out from S1 to S2, also shifts the investment schedule, and, as drawn, reduces the equilibrium interest rate in the market for loanable funds:

"As Hicks told us -- and as Keynes himself told us in Chapter 14 -- what the supply and demand for funds really give us is a schedule telling us what the level of income will be given the rate of interest. That is, it gives us the IS curve of Figure 5, which tells us where the central bank must set the interest rate so as to achieve a given level of output and employment. Of course, as the figure indicates, it's possible that the interest rate required to achieve full employment is negative, in which case monetary policy is up against the zero lower bound, that is, we're in a liquidity trap. That's where America and Britain were in the 1930s -- and we're back there again.

"One way to think about this situation is to draw the supply and demand for loanable funds that would prevail if we were at full employment, as in Figure 6. The point then is that there's an excess supply of desired savings at the zero interest rate that's the lowest achievable. A zero-lower-bound economy is, fundamentally, an economy suffering from an excess of desired saving over desired investment.

I shall begin by making a few peripheral comments. Keynes says a lot about saving and investment (as does Krugman in all of the above). At full employment saving equals investment. When saving exceeds investment substantially you get a recession. But why is it that Keynes (and, by extension, Krugman), except for the very rare casual reference to over-investment, never considers it possible that investment can run over saving for a considerable length of time? The answer is that such a possibility is impossible in a world where money consists entirely of currency. If the saver has $100 the investor can use that $100 at full employment. If the saver decides to hold on to part of the $100 or the investor decides to invest less than $100 you get a recession. But as long as $100 is currency it is impossible for the investor to invest more than $100. That is possible only when banks are introduced. And since there are no banks in Keynes the possibility never occurs to him or to Krugman.

Now it is clear why all through the period from 2002 to the beginning of 2006 no warning bells struck in Krugman's mind. The idea that investment can run higher than saving is an impossibility in the Keynesian scheme of things. Now Keynesians will object that if investment was running over saving for such an extended period why is it that there was no raging inflation during this period. By the way, there was no raging inflation before the 1929 crash either. After all, if factor payments were made while making the investment and no corresponding consumer goods were manufactured, inflation should have been the natural result.

It is not surprising that Keynesians should think in this fashion. For the counterpart to the fact that in the Keynesian model there are no banks is the fact that there are no financial assets either. I can visualise Keynesians rising to object that Keynes made extensive references to beauty contests, casinos, the bubble on a whirlpool of speculation and so on. All true, but that does not change the fact that there are no financial assets in Keynes.

This is what Keynes said in Chapter 7:

"So far as I know, everyone agrees in meaning by Saving the excess of income over what is spent on consumption. It would certainly be very inconvenient and misleading not to mean this. Nor is there any important difference of opinion as to what is meant by expenditure on consumption."

The assumption, of course, is that there is no expenditure on financial assets. When the latter are introduced, the equation changes completely. Saving is then not the difference between income and consumption. Saving is the difference between income and the sum of the expenditure on consumption and the expenditure on financial assets.

It is not that between 2002 and 2006 investment was running far above its long-term trend. It was that saving was running far below its long-term trend, and the higher expenditure did not make itself felt as inflation because it was being spent on financial assets. The same thing was happening in 2010 and the first quarter of 2011 but as during 2002-2006 Krugman cannot understand it because in his economic model there are no banks and no financial assets. [In all of the above I am assuming a closed system; in an open system the higher expenditure can be on imported goods as well.]

So, in clamouring to keep interest rates low after the 2001 recession, Krugman brought on the Great Recession, which his friend Brad deLong now wants to call the Little Depression, and in clamouring for low interest rates since 2008, Krugman is bringing on the next recession.

I can imagine an entry in an economics encyclopaedia a hundred years from now:

Krugmanian: Term used to describe a long and painful recession that follows bank failures after a financial crash. Typically this is the result of holding interest rates low for a long period, often as a way of escaping a previous recession. So named after Paul Krugman, an economist who cheer-led two successive recessions in the first two decades of the 21st century. Krugman, although not a business cycle theorist, rose to prominence as a vocal exponent of the ideas of J.M. Keynes, an economist whose ideas held sway in the second half of the 20th century, but is now regarded as being of minor, mainly historical, interest.

After those peripheral comments, it's time to begin a serious mathematical assault on Krugman. I must say at the outset that although Krugman may be good at drawing up utility maximising functions his knowledge of elementary calculus is quite poor. Fighting words, those, but I am going to show why.

To quote Krugman:

"As Hicks told us -- and as Keynes himself told us in Chapter 14 -- what the supply and demand for funds really give us is a schedule telling us what the level of income will be given the rate of interest. That is, it gives us the IS curve of Figure 5, which tells us where the central bank must set the interest rate so as to achieve a given level of output and employment."

In other words, the schedule gives us a function relating income, or GDP, to the rate of interest r. Now what does it mean to say in calculus that y is a function of x. It means that to every x in our domain of interest, there corresponds one and only y. So to every interest rate there corresponds one and only one value of GDP.

But it is very easy to show that this cannot be. For instance, a low interest rate could mean that the demand for money is low. But it could also mean that the demand for money is high and is being accommodated by a high supply of money. So from the interest rate alone it is impossible to determine the demand for and supply of money, any more than it is possible to determine from the price of potatoes alone the demand for and supply of potatoes. And that is not all. Given a certain amount of money it is impossible to determine uniquely the value of GDP. For instance, an expenditure of 1000 billion dollars could be partitioned as 900 billion dollars on real goods and services (ie on GDP) and 100 billion dollars on financial assets. Or it could be 800 billion dollars on GDP and 200 billion dollars on financial assets. There are an infinite number of ways in which the partitioning can take place.

In other words, given a certain interest rate it is impossible to determine the value of GDP from it. Income is not a function of r, and therefore every one of Krugman's (and Keynes's) assertions above, including Figs 4, 5, and 6 and much else are impossibilities. The assumption of course is that money is constant, which is to say, that money consists entirely of currency and there are no banks and no financial assets.

Krugman wonders why so many PhD economists find it hard to accept so many Keynesian assertions. He can think of only two possible reasons. One, the economists are dim-witted. Two, they are consciously or unconsciously acting on behalf of the rich. The third possibility never seems to have occurred to him: that Keynes and he are wrong.

A lot of economists have intuitively realised that there was something about the General Theory that did not quite add up. But they were not able to nail the reasons, because they themselves had internalised many of its assumptions. I wonder whether Krugman himself will have the guts to abandon Keynesianism. I think not. It takes a lot of courage to admit that you've been making a fool of yourself for more than a decade. By the way, Mr Krugman, if you feel upto examining new theories, here is a model of the economy incorporating banks and endogenous money: The Riddle of Money, Finally Solved

Methinks a 100th anniversary conference to commemorate the General Theory will not be held.